Theorem pnrmtop 23262

Description: A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
pnrmtop	⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Proof of Theorem pnrmtop

Step	Hyp	Ref	Expression
1		pnrmnrm 23261	. 2 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Nrm)
2		nrmtop 23257	. 2 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
3	1, 2	syl 17	1 ⊢ (𝐽 ∈ PNrm → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2111  Topctop 22814  Nrmcnrm 23231  PNrmcpnrm 23233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-cnv 5627  df-dm 5629  df-rn 5630  df-iota 6443  df-fv 6495  df-ov 7355  df-nrm 23238  df-pnrm 23240

This theorem is referenced by:  pnrmopn  23264